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A349394 a(p^e) = p^(e-1) for prime powers, a(n) = 0 for all other n; Dirichlet convolution of A003415 (arithmetic derivative of n) with A055615 (Dirichlet inverse of n).

%I A349394 #51 Mar 03 2023 17:11:28
%S A349394 0,1,1,2,1,0,1,4,3,0,1,0,1,0,0,8,1,0,1,0,0,0,1,0,5,0,9,0,1,0,1,16,0,0,
%T A349394 0,0,1,0,0,0,1,0,1,0,0,0,1,0,7,0,0,0,1,0,0,0,0,0,1,0,1,0,0,32,0,0,1,0,
%U A349394 0,0,1,0,1,0,0,0,0,0,1,0,27,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,1,0,1,0,0
%N A349394 a(p^e) = p^(e-1) for prime powers, a(n) = 0 for all other n; Dirichlet convolution of A003415 (arithmetic derivative of n) with A055615 (Dirichlet inverse of n).
%C A349394 Dirichlet convolution of this sequence with Euler phi (A000010) is A300251.
%C A349394 Convolving this sequence with sigma (A000203) produces A319684.
%C A349394 With a(1) = 1 instead of 0, this would be the Dirichlet convolution of A129283 (A003415(n)+n) with A055615. Thus when we subtract A063524 from that convolution, we get this sequence. (See also A349434). Compare also to the convolution of A069359 (sequence agreeing with A003415 on squarefree numbers) with A055615, which is the characteristic function of primes, A010051. - _Antti Karttunen_, Nov 20 2021
%H A349394 Antti Karttunen, <a href="/A349394/b349394.txt">Table of n, a(n) for n = 1..20000</a>
%H A349394 P. Haukkanen, J. K. Merikoski and T. Tossavainen, <a href="http://www.mathos.unios.hr/mc/index.php/mc/article/view/3206">Asymptotics of partial sums of the Dirichlet series of the arithmetic derivative</a>, Mathematical Communications 25 (2020), 107-115.
%F A349394 a(n) = Sum_{d|n} A003415(n/d) * A055615(d).
%F A349394 a(n) = 0 unless n is a prime power (A246655), in which case a(p^e) = p^(e-1). - _Sebastian Karlsson_, Nov 19 2021
%F A349394 a(n) = A003557(n) * A069513(n). [From above] - _Antti Karttunen_, Nov 20 2021
%F A349394 Dirichlet g.f.: Sum_{p prime} 1/(p^s-p) [Follows from the D.g.f. of A003415 proved by Haukkanen et al.]. - _Sebastian Karlsson_, Nov 25 2021
%F A349394 Sum_{k=1..n} a(k) has an average value c*n, where c = A137245 = Sum_{primes p} 1/(p*log(p)) = 1.63661632335... - _Vaclav Kotesovec_, Mar 03 2023
%t A349394 f[p_, e_] := e/p; d[1] = 0; d[n_] := n * Plus @@ f @@@ FactorInteger[n]; a[n_] := DivisorSum[n, # * MoebiusMu[#] * d[n/#] &]; Array[a, 100] (* _Amiram Eldar_, Nov 19 2021 *)
%o A349394 (PARI)
%o A349394 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A349394 A055615(n) = (n*moebius(n));
%o A349394 A349394(n) = sumdiv(n,d,A003415(n/d)*A055615(d));
%o A349394 (PARI) A349394(n) = { my(p=0,e); if((e=isprimepower(n,&p)),p^(e-1),0); }; \\ (After _Sebastian Karlsson_'s new formula) - _Antti Karttunen_, Nov 20 2021
%o A349394 (Haskell)
%o A349394 import Math.NumberTheory.Primes
%o A349394 a n = case factorise n of
%o A349394     [(p,e)] -> unPrime p^(e-1) :: Int
%o A349394     _ -> 0 -- _Sebastian Karlsson_, Nov 19 2021
%Y A349394 Cf. A003415, A003557, A055615, A063524, A069513, A100995, A120007, A129283, A246655.
%Y A349394 Cf. also A000010, A000203, A069359, A300251, A319684, A327564, A349340, A349396, A349434, A349618, A349619, A349620, A349621.
%K A349394 nonn
%O A349394 1,4
%A A349394 _Antti Karttunen_, Nov 18 2021
%E A349394 Added _Sebastian Karlsson_'s formula as the new primary definition - _Antti Karttunen_, Nov 20 2021